1. Dragonball or any of the Big 3. All I'm really saying here is that a long-running shounen should be required. There are so many parodies, plot devices, and themes in this slot that, in order to understand them, you need to know how these shows work.

2. Rurouni Kenshin OVA's, Cowboy Bebop, Fullmetal Alchemist, Spirited Away, Legend of the Galactic Heroes etc. Here, this gives you an idea, whether you like the works or not, of what the "community" generally chooses as the representatives of "quality" in anime.

3. FLCL, Gurren Lagann, Redline, Bakemonogatari, Kyousougiga, Tatami Galaxy, Revolutionary Girl Utena, Mawaru Penguindrum etc. Many, even most, anime (given the right staff) could be just as good in another medium. These, however, would have a tough time in live-action, or at the very least, would feel like a completely different work. Watch a couple of these, and you'll understand a lot of things that make animation rather unique from live-action shows, novels, manga, etc.

Edit: Also, I wouldn't say any of these are "required," but rather that they are good examples of what "anime" is, and that watching these will give you a good idea of what people mean when they talk about "anime." I would also add in harems and slice-of-life's as categories, but really, you can probably pick a couple of shows a season that would serve as excellent examples.

Sometimes I hate not having an original idea, but I have to put the vote out there for:

#1 Tengen Toppa Gurren Lagann

"Required viewing" should be something that encompasses a genre while remaining accessible to outsiders or first-timers. TTGL has everything from beautiful leads to tragic events, yet retains the meta-humor on a level rarely matched.

Well, I don't really believe in "required viewing" because it seems biased toward what someone personally likes. If I had to pick required series, I'd probably go with

The Melancholy of Haruhi SuzumiyaDeath NoteToradora (Possibly replace this with Anohana)Clannad and Clannad ~ After Story ~

Now, as I said, I don't really believe in required viewing. It's not like these series were so awesome that everyone must see them. Rather, these are series that everyone and their grandmother seems to talk about all the time so I think some familiarity with them is a good idea.

EDIT: If your a yuri fan, i'd say

Maria-sama ga Miteru
Strawberry Panic
Kannazuki no Miko (Though for the life of me, I don't know why. I literally watched it only because it was required).
And possibly Revolutionary Girl Utena

Full Metal Panic
Hikaru No Go
Stein;s Gate
Death Note
Black Lagoon
Mushishi
Everything by Miyazaki and Makoto Shinkai
Time of Eve
Usagi Drop
Bakemonogatari, etc.
Anohana
Chihayafuru
My Ordinary Life
Cross Game
Eden of the East
The Devil is a Part Timer
My Teen Romantic Comedy, SNAFU
Level-E

I'd be down if anyone had a suggestion to one of those genres with a deeper plot besides "alien/magical girl/boy shows up on prepubescent girl's/boy's doorstep."

At that point, it ceases to be "slice of life." I would suggest His and Her Circumstances (Kare Kano) for a romance that is grounded, but still heartwarming and hilarious.

One type of anime that i've actually been asked is if there is an oldschool horror anime. Not like Gantz where it's just (mild spoiler) brutal and a you're fucked kind of thing. But like an old maybe even slendermanish horror. You know, run for your life, mcguyver some shit or your dead kinda thing. I've yet to come across one.

I think Boogiepop Phantom is the only thing that comes to mind and comes close to hitting those bullet points.

For robots or war: Mobile Suit Gundam / Space Runaway Ideon (same director, but right after the first season of Gundam. Ideon influenced the next few Gundam Series. Much of Evangelion seems to be based off of Ideon in some way)

Cyberpunk: Akira (movie) (There's a lot that Akira perfected together, and the themes were the standard for a while)

Shakugan no Shana set a large number of standards for "little angry protagonist girls"

Le Portrait de Petit Cossette (movie): The director's anime & movies have a weird use of colors and overlapping textures and B-role that were basically perfected by the time this movie was made. There are SO MANY anime that have been influenced by this, that it's kind of silly not to mention one of the director's earlier works. (See: The Soul Taker for his "experimental" phase)

Fist of the North Star did quite a lot for action anime.
However, as much as I hate it, Dragon Ball Z set a lot of standards for your typical"boy's action show" (largely subverting Fist of the North Star.) The original Dragon Ball didn't do much that Dr. Slump or Astro Boy didn't already do.

The Ghibli collection.
Ghost in the Shell first movie and SAC series 1 and 2 then watch Solid State Society movie
One Piece,Dragonball, Fist Of The North Star,Full Metal Alchemist and all that Shonen goodness
Cowboy Bebop
Captain Harlock,Galaxy Express 999 and anything from that universe
Akira,Ninja Scroll,Bubblegum Crisis,Dominion Tank Police,Genocyber,Battle Angel Alita and many more classic 80's anime
Serial Experiments: Lain. Probably the most thought provoking and deep anime ever made.
Most of the Gundam series is great and worth watching if you love mecha anime. Also check out Evangelion and Eureka Seven
FLCL
Melancoly of Haruhi Suzmiya
Welcome To The NHK
Blackjack